It seems like the reward will always have to be > burn otherwise no one would do it so burning is always net-positive meaning everyone wants to do it as fast as possible to claim the rewards.
It is true that the "expected" total reward (for a participant) must be higher than the burnt amount of coins. If the "total burnt amount" is very high, it can lead to a situation where it is not profitable to burn coins in this moment. In a mature market it can be expected that this will lead to an equilibrium where there will never be "too much" nor "too few" coins burnt.
This equilibrium should lead to a situation where burning it is not guaranteed to be profitable, but with a certain risk it can lead to profit - but in a certain amount of time, e.g. in some months. That means that a "burnt coin" can be considered a long-time investment in the coin. My theory is that this mechanism can stabilize the coin value a bit as these long term investors cannot sell their burnt coins (they can sell their private keys, but in a trustless environment this is very difficult).
If everyone in the network was participating in burning equally, everyone is getting an equal reward at the expense of everyone else, do we need a reward system at all? Everyone is getting the same reward.
This situation is unlikely to happen, as for this there must be a very even distribution of coins.
If a subset of people in the network are participating in the burning then that subset is getting a reward at the expense of everyone not in that subset, is this desirable? Since there's no physical limitation to burning it seems like in the end the person with the largest balance will get the largest reward in the long run.
Yes, that is true and that is basically the same "problem" like in Proof of Stake, ("rich get richer") but with an important difference. Burnt coins can not be moved (and spent) inmediatly because they are destroyed when burnt; you only can spend your profit after the rewards have been paid. In the situation I describe above, it will last several months to get a profit when PoB mining. This means that a "burning miner" is locking coins for some months.
So the game is a little bit different compared to PoS: it's not
wealth but
risk what is rewarded, because the value of a PoB currency can drop during these months.
@monsterer: Thank you for your analysis. I will respond in your thread after I've read more material regarding the attacks you're mentioning. A quick question: Are the attacks you're describing also possible for a pure Proof of Stake system (and so comparable to the "nothing at stake" problem)?